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Chapter 2 Fundamentals of Tissue Optics
Chapter 2. Fundamentals of Tissue Optics
7
2.1 Introduction
The application of lasers and other optical technology to problems in
biomedicine is a rapidly growing field. These applications can be classified as
either therapeutic or diagnostic. In therapeutic applications, the transformation
of light energy into chemical, thermal or mechanical energy, via light
absorption, can cause a direct and selective cell death (Niemz, 1999). In this
project the fluence rates are chosen to be sufficiently small, thus these effects
can be ignored. The light-tissue interactions in the diagnostic approach must
by contrast be non-destructive and the main goal is to study the physiology or
pathology of the tissue. There are a variety of potential optical methods to
evaluate the light-tissue interactions, such as diffuse reflection spectroscopy
and time resolved transmittance among others.
The fundamental optical characteristics that are exploited for diagnostic
information are absorption and scattering (elastic and inelastic) in the
wavelength region of 600 – 1000nm (near-infrared, NIR range) where tissue
scattering predominates over absorption. This research project concerns the
use of sources in the near-infrared range only.
2.2 Optical properties
Photon propagation in biological tissue is characterised by the basic optical
properties of absorption, scattering, and refractive index. These properties
govern the numbers of photons that are transmitted between points on the
surface of tissue.
2.2.1 Refractive Index
The simplest of the optical properties of tissues is the refractive index n, which
determines the speed of light in the medium. Changes in the refractive index,
either continuous or abrupt (at boundaries), give rise to scattering, refraction
and reflection. Refraction usually occurs when light is incident at the boundary
between two media of different indices of refraction (Figure 2.1),
Chapter 2. Fundamentals of Tissue Optics
8
Figure 2.1 Refraction of light between two media with different refractive index (n1<n2).
and is governing by Snell’s law. It states that:
)sin(.v)sin(.v 2211 θθ = [2.1]
where θ1 is the angle of incidence, θ2 is the angle of refraction (to the normal
of the interface), and ν1 and ν2 are the speeds of light in the media, and are
related to the refractive index of each media (n = cvacuum/ν). Since tissues are
heterogeneous in composition, one may need to know the refractive indices
for the various tissue constituents or an averaged value for the tissue as a
whole. The overall refractive index is considered to be around 1.4 for most
tissue types (Delpy et al, 1988).
2.2.2 Absorption
The transmitted light intensity I(d) across a homogeneous and non-scattering
medium of thickness d, which is illuminated by a collimated beam of light of
intensity I0 at the wavelength λ will be given by: )d.(
)d(ae.II µ−= 0 [2.2]
Figure 2.2 Attenuation of light through a non-scattering medium.
where µa is the absorption coefficient of medium [mm-1] for a given
wavelength λ, and it represents a probability per unit length of a photon being
absorbed.
µa I(d)
I0
d
n2
n1
θ2 θ1
ν1
ν2
Chapter 2. Fundamentals of Tissue Optics
9
Other parameters are also defined from the ratio of the transmitted to incident
intensity, such as the Transmittance T, which is usually what a spectroscopic
instrument measures:
0I/IT )d(= [2.3]
and Absorbance A (representing the loss in light intensity), which is usually
measured in units of optical density (OD), and is given by equation [2.4].
( ) d.kIIlogTlogA
)d(=
== 0
10101
[2.4]
An equivalent absorbance relationship was developed by Beer in 1852
(equation [2.5]). It states that for an absorbing compound dissolved in a non-
absorbing medium, the attenuation is proportional to the concentration of the
compound in the solution ([C], [molar]), the specific extinction coefficient of the
compound (ε, [molar-1.mm-1]) and distance between the points where the light
enters and leaves the solution (the optical pathlength d).
===
)d(IIlogd.kd].C.[A 0
10ε [2.5]
The equation [2.5] can be extended if the solution contains several different
absorbing compounds, considering the contributions of each compound (for
instance, components of blood like oxy and deoxy- haemoglobin, water, etc):
=++= d].C).[(...d].C).[(d].C).[()(A nn λελελελ 2211 d.]C).[(
nnn∑= λε
[2.6]
Rewriting the equation [2.6] using the natural logarithm base yields:
d].C).[(...d].C).[(d].C).[(IIlog nn
)d(e λαλαλα ++=
22110
d.]C).[(n
nn∑= λα
[2.7]
where α is the specific absorption coefficient of the compound
([molar-1.mm-1]). This differs from the specific extinction coefficient (ε) by a
scaling factor equal to loge(10).
Chapter 2. Fundamentals of Tissue Optics
10
2.2.3 Scattering
Scattering is a physical process by which light interacts with matter to change
its direction, so if the medium is scattering, the path taken by the photons are
no longer direct (see section 2.6.1). Light scattering in tissue depends upon
many variables including the size of the scattering particle, the wavelength of
the light and the variation of the refractive indices of the various tissue
components, such as cell membranes and organelles.
Figure 2.3 Attenuation of light through a scattering medium.
Elastic scattering (i.e. no loss of energy) can still give rise to attenuation of a
light beam by deflecting photons from their initial path. In the same manner as
for absorption, the final non-scattered intensity component of light I(d),
transmitted through a medium of thickness d when illuminated by a source of
intensity I0 is described by: )d.(
)d(se.II µ−= 0 [2.8]
where µs is the scattering coefficient of the medium [mm-1] for a given
wavelength λ, and represents a probability per unit of length of a photon being
scattered.
2.2.4 Anisotropy and the Coefficient of Anisotropy g
The practical effect of scattering and absorption by a particle upon a parallel
beam of light propagating in one given direction is that the beam intensity in
this direction is reduced. Light which is absorbed is dissipated as thermal
energy, and light which is scattered keeps its intensity but travels in another
direction (Cope, 1991). Therefore, it is convenient to describe the angular
I0
µs
I(d)
d
Chapter 2. Fundamentals of Tissue Optics
11
distribution of scattered light by defining an angular probability function of a
photon to be scattered by an angleθ. If all scatter directions are equally
probable, the scattering is isotropic. Otherwise, anisotropic scattering occurs.
In elastic scattering, when a photon is scattered from its original direction (s)
by a particle, it emerges in a new direction (s’), as shown in Figure 2.4. The
angular probability of this change in direction is given by the phase function
p(s,s’) over its domain (solid angle Ω of 4π steradians).
Figure 2.4 Elastic scattering event, based on (Vo-Dinh, 2003) (p2-7).
For a random medium that is isotropic in terms of its physical properties
(refractive index, density), it can be assumed that this probability is
independent of direction s and only depends on the angle between the
incident and scattered directionsθ. Thus, the phase function can be expressed
as a function of the scalar product of the unit vectors in the initial and final
directions, which is equal to the cosine of the scattering angle θ:
( ))cos()'.()',( θ== psspssp [2.9]
A measure of the anisotropy of scattering is given by the coefficient of
anisotropy g, which represents the average value of the cosine of the
scattering angle. This can be expressed as:
( ) ( )∫−
=1
1
)cos(d.)cos(p).cos(g θθθ [2.10]
The value of g approaching 1, 0, and -1 describe extremely forward, isotropic,
and highly backward scattering respectively, and in biological tissue, g lies in
the range 0.69 ≤ g ≤ 0.99 (Cheong et al, 1990).
INCIDENT PHOTON
SCATTERED PHOTON
θ
SCATTERER Direction S’
Direction S cos(θ)
Differential solid angle
dΩ
Chapter 2. Fundamentals of Tissue Optics
12
It is also convenient to express the characteristic scatter of tissue in terms of
the transport scatter coefficient (µs’), which represents the effective equivalent
number of isotropic scatters per unit of length, usually [mm-1], and is used in
the diffusion theory of light propagation in random media. Thus,
)g.(' ss −= 1µµ [2.11]
Finally, the total transport attenuation coefficient (µtotal) can be found:
asastransport_total )g.(' µµµµµ +−=+= 1 [2.12]
2.3 Absorption characteristics of the main chromophores in tissue
The tissue compounds which absorb light in the spectral region of interest are
known as chromophores. Each chromophore has its own particular absorption
spectrum which describes the level of absorption at each wavelength. In the
near-infrared range (NIR), known as the absorption window or therapeutic
window, the major absorbing components in the soft tissues are water,
oxyhaemoglobin and deoxyhaemoglobin. There are also minor contributions
from other tissue chromophores, such as melanin, lipids, etc. The absorption
spectra of some common chromophores are shown in figure 2.5.
Figure 2.5 The absorption spectra for the main chromophores found within tissue.
At the shorter wavelength end, the window is bound by the absorption of
haemoglobin (in both its oxygenated and deoxygenated forms). At the IR end
of the window, penetration of light is limited by the absorption properties of
water. Within the therapeutic window, scattering is dominant over absorption,
and the propagation of light becomes diffuse (Niemz, 1999), (Elwell, 1995).
Chapter 2. Fundamentals of Tissue Optics
13
The concentration of water and melanin remains virtually constant with time
(static absorbers). On the other hand, the concentrations of dynamic
absorbers, such as oxygenated and deoxygenated haemoglobin (related with
blood oxygenation), and cytochrome oxidase (an enzyme in the oxidative
metabolic pathway that provides an indicator of tissue oxygenation and cell
metabolism), provide clinically useful physiological information. However, the
concentration of cytochrome oxidase in tissue is inferior when compared with
haemoglobin (at least one order of magnitude below that of haemoglobin)
(Elwell, 1995), (Cope and Delpy, 1988).
2.3.1 Water
The average water content of the neonatal brain is 90% (Cope, 1991) and in
adult brain is about 80% of its weight (Woodard and White, 1986). Because of
its high concentration in most biological tissue, water is considered to be one
of the most important chromophores in tissue spectroscopy measurements.
Nevertheless, for the purposes of most clinical measurements the water
concentration in tissue can be thought of as constant, and as such water acts
as a fixed constant absorber (Elwell, 1995). From figure 2.6, it can be seen
that water has a low absorption over the NIR range (µa = 0.0022 mm-1 at
800 nm). Beyond 900 nm absorption rises sharply with increasing wavelength,
with a spectral peak being visible at 970 nm and this sets an upper limit for
spectroscopic or imaging measurements.
Absorption Curve of Water
Wavelength (nm)
650 700 750 800 850 900 950 1000 1050
Abs
orpt
ion
Coe
ffici
ent (
mm
-1), µ a
0.000
0.010
0.020
0.030
0.040
0.050
0.060
Figure 2.6 The Absorption spectrum for pure water at 37oC over the wavelength range
from 650 –1050 nm (Matcher et al, 1993).
Chapter 2. Fundamentals of Tissue Optics
14
2.3.2 Lipids (fat)
The absorption spectrum of lipids is similar to that of water. However, its
overall contribution to absorption is relatively small due to the low content of
fat in the brain (about 5% of the total net weight of a newborn infant’s brain)
(Cope, 1991). The lipids are considered a fixed constant absorber with
concentration normally unchanging during clinical measurements and so the
measurements of changes in attenuation are not affected (Elwell, 1995).
Figure 2.7 shows the absorption spectrum of pure pork fat between 650 nm
and 1000 nm, which is thought to be similar to that of human lipids in muscle
tissue (Conway et al, 1984).
Absorption Curve of Lipids
Wavelength (nm)
650 700 750 800 850 900 950 1000
Abs
orpt
ion
Coe
ffici
ent (
m-1
), µ a
0
2
4
6
8
10
12
14
Figure 2.7 Spectrum of pork fat in the NIR from 650 to 1000 nm (van Veen et al, 2000).
2.3.3 Haemoglobin
The average composition by volume of blood is: 54% of plasma, 45% of red
blood cells and 1% of white blood cells and platelets in the normal adult
(Marieb and Hoehn, 2006). Haemoglobin molecules, which can be found
within the red blood cells, carry 97% of the oxygen in the blood, while 3% is
dissolved in the plasma. Each haemoglobin molecule consists of the four
haem groups (iron atom at the centre of its structure, which has certain
paramagnetic properties, see section 3.6) bound to the protein globin (figure
2.8) (Martini et al, 1998). Haemoglobin has an important role in the transport
and delivering of oxygen from the lungs to tissues (oxyhaemoglobin), and
carrying carbon dioxide from the tissue (deoxyhaemoglobin) back to the lungs.
Chapter 2. Fundamentals of Tissue Optics
15
Figure 2.8 The structure of Haemoglobin, which consists of four globular protein subunits
(α and β chains), and each subunit contains a single molecule of haem, a porphyrins ring surrounding a single ion of iron, (Martini et al, 1998) (p630).
The amount of haemoglobin in the blood determines how much oxygen the
red blood cells are capable of carrying to other cells. The normal ranges for
haemoglobin concentrations are (which change according age and sex) 14 to
20 grams per decilitre in infants, 13 to 18 g/dL in adult males, and 12 to
16 g/dL in adult females (Marieb and Hoehn, 2006). Haemoglobin molecules
(Hb) in the red blood cells are responsible for almost all of the absorption of
light by blood. However, the absorption spectrum of haemoglobin (figure 2.9)
changes when oxygenation/de-oxygenation occurs. Oxygenated haemoglobin
is a strong absorber up to 600 nm (sets a lower limit for spectroscopic or
imaging measurements); then its absorption drops off very sharply and
remains low. The absorption of deoxygenated haemoglobin, however, does
not drop sharply; it stays relatively high, although it decreases with increasing
wavelengths. The two absorption spectra cross around 800 nm (the isosbestic
point, αHbO2 = αHb).
Absorption Curves of Haemoglobin
Wavelength (nm)
650 700 750 800 850 900 950 1000
Spec
ific
Abs
orpt
ion
Coe
ffici
ent (
mm
-1.(µ
Mol
)-1), α
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
Deoxyhaemoglobin
Oxyhaemoglobin
Isosbestic point
Figure 2.9 Specific absorption coefficients (α) of the states of haemoglobin (Cope, 1991).
Chapter 2. Fundamentals of Tissue Optics
16
In addition, foetal haemoglobin (HbF) has higher affinity to oxygen than adult
haemoglobin (HbA). This facilitates the uptake of oxygen by the foetus from
the mother’s blood by placenta. The visible absorption spectra of HbF and
HbA are slightly different but they are virtually identical in the NIR range
(figure 2.10). Zijlstra et al (1991) studied these differences and determined the
specific extinction coefficients (ε) for enriched and reduced states of HbF and
HbA in the range of 450-1000 nm. He showed in his calculations of blood
oxygen saturation (SO2) that a method based on 2 wavelengths, such as
pulse oximetry (660/940 nm), can result in an underestimation of SO2 if HbF
instead of HbA is being measured. However, the studies of Wickramasinghe
et al (1993) have shown that adult Hb can be assumed for infants without
introducing a significant error.
Absorption Curves of Haemoglobin for Adult and Foetus
Wavelength (nm)
700 750 800Spec
ific
Extin
ctio
n C
oeffi
cien
t (cm
-1.(m
Mol
)-1), ε
0.0
1.0
2.0
Foetal Hb Adult HbFoetal HbO2 Adult HbO2
Figure 2.10 Spectra of adult and foetal absorption curve (Zijlstra et al, 1991). 2.4 The origin of optical contrast in the human brain
The delicate tissues of the brain, which contains tens of billions of neurons
arranged in a 3D structure and performs a complex array of functions, are
protected and surrounded by the skin, the flat bones of the cranium, the
cranial meninges (dura mater, arachnoid mater and pia mater), and the watery
fluid cerebrospinal fluid (CSF) (figure 2.11).
Chapter 2. Fundamentals of Tissue Optics
17
Figure 2.11 Human brain and surrounding structures (Martini et al, 1998) (p441).
In applications involving NIR spectroscopy of the brain, light must pass
through the skull and surface tissues layers before entering and exiting the
brain. Firbank et al (1993) stated the optical properties of these tissues and
surrounding structures must be known in order to model the effects of the
tissue layers and determine chromophores concentrations. Some of the
optical characteristics of these biological tissues will be discussed in the
following sections.
2.4.1 Skin
Skin covers the entire surface of the body and protects internal organs from
the harsh elements of the environment. It also protects the body from injury by
acting as a shock absorber (figure 2.12).
Figure 2.12 Skin and underlying subcutaneous tissue (Martini et al, 1998) (p199).
Chapter 2. Fundamentals of Tissue Optics
18
The main component layers of the skin are:
• the epidermis, which is the outermost layer of the skin, and is composed of
millions of dead skin cells, has no blood vessels and has the natural
protective pigment responsible for skin colour, known as melanin;
• the underlying connective tissue of the dermis, which is compose of nerves
and is vascularised, with sweat glands and hair follicles, and
• beneath the dermis, the hypodermis is a subcutaneous adipose tissue.
Due to melanin in the epidermis layer, light is highly absorbed, as shown in
table 2.1, and especially in the ultraviolet region (Elwell, 1995). Simpson et al
(1998) concluded that transmission of light through the skin will be highly
dependent on the pigmentation of the skin.
Table 2.1 Optical coefficients of human dermis, (Simpson et al, 1998).
SKIN LAYER µa [mm-1] µs’[mm-1] DERMIS + EPIDERMIS CAUCASIAN 0.033 ± 0.009 2.73 ± 0.54
HYPODERMIS CAUCASIAN 0.013 ± 0.005 1.26 ± 0.34 DERMIS + EPIDERMIS NEGROID 0.241 ± 0.153 3.21 ± 2.04
MEASUREMENTS REALIZED EX VIVO AT 663 nm.
It is noticeable from the values shown in table 2.1 that the attenuation of light
by the skin is strongly dominated by scattering. Simpson et al (1998) also
concluded that for both dermis and epidermis the transport scattering
coefficient decreases monotonically, with increasing wavelength (i.e. a
general decrease of scatter due to the size of the particle in comparison with
the wavelength).
2.4.2 Bones & Skull
The skull grows and expands in proportion to the growth of the brain. At (term)
birth, the majority of the bones of the skull are ossified (i.e. the bones have
developed from dense connective tissue and cartilage). The areas where the
bones join together are called sutures, and where they have not come
together is covered by areas of fibrous connective tissue known as fontanelles
(figure 2.13).
Chapter 2. Fundamentals of Tissue Optics
19
Figure 2.13 The skull of a term neonate, extracted from (Martini et al, 1998) (p211).
These connections are quite flexible enabling distortion without damage to the
skull, and ease the passage of the infant through the birth canal. Damage
could occur if pressure is applied to the head. In addition, the skull is a smooth
bone, made up of two thin layers of compact bone (responsible for the
skeleton's strength) compressing an irregular layer of spongy bone (made up
of a network of tiny strands of bone called trabeculae), which contains red
bone marrow (diploë) (figure 2.14).
Figure 2.14 Cross section of flat bone of skull (Martini et al, 1998) (p170).
Firbank et al (1993) have studied the optical properties of adult skull in the
wavelength range 650 to 950 nm, obtaining values of µa = 0.04 to 0.05 mm-1
and µs’ = 2.7 to 1.3 mm-1. The average value of g varied from 0.925 to 0.945.
The presence of a small quantity of blood in bone has little effect on the
overall absorption by the bone (Elwell, 1995).
2.4.3 Cerebrospinal fluid and membranes
Cerebrospinal fluid (CSF) and membranes surround the brain and much of the
spinal cord. CSF circulates through the cerebral ventricles (internal structures
in the brain, formed by four chambers), supporting, cushioning and sustaining
Chapter 2. Fundamentals of Tissue Optics
20
the brain (Martini et al, 1998). CSF is optically very similar to water (i.e. CSF is
non-scattering) with an absorption coefficient (µa) of 0.0022 mm-1 at 800 nm.
Some membranes that surround the brain are thin, delicate and highly
vascularised (pia matter), and others are tough and fibrous, forming a sack
around the brain (dura matter). The scattering properties of each membrane
depend on its structure, although the membranes are very thin. The
vascularised membranes will absorb according to the amount of blood
present. The membranes themselves are low absorbing (Hillman, 2002). Very
few models of light propagation in the adult and neonatal heads take into
account the presence of CSF fluid and the surrounding membranes.
2.4.4 Characteristics of the human brain
2.4.4.1 Neurons and Cerebral cortex
Nerve tissue in the brain has two cell types: neurones (or nerve cells) and
supporting cells knows as neuroglia. The number of neurons in the human
body is estimated to be 200 billions and around half them can be found in the
brain. The neurones with their nerve cell bodies (soma) and their axons (figure
2.15) are the main components of the grey and white mater in the brain.
Figure 2.15 A neurone or nerve cell, extracted from (Martini et al, 1998) (p134).
The cerebral cortex constitutes a superficial layer of grey matter (high
proportion of nerve cell bodies) and internally the white matter which is
responsible for communication between axons. The white matter appears
white because of the multiple layers formed by the myelin sheaths around the
axons, which are the origin of the high, inhomogeneous and anisotropic
scattering properties of brain (figure 2.16).
Chapter 2. Fundamentals of Tissue Optics
21
Figure 2.16 Representation of cross section of the adult brain, showing the grey and
white matter.
Extracted from Rolfe (2000) and van der Zee (1992), table 2.2 shows values
of optical properties of brain tissue for the adult and neonatal brain.
Table 2.2 Optical coefficients of human dermis, (Simpson et al, 1998).
PARAMETER BRAIN TISSUE ADULT BRAIN
NEONATAL BRAIN [40 WEEK GESTATION]
WHITE MATTER 0.957* 0.982* g GREY MATTER 0.82* 0.978* WHITE MATTER 0.0032 – 0.01 0.037 – 0.048 µa [mm-1] GREY MATTER 0.032 – 0.038 0.033 – 0.05 WHITE MATTER 9.26 – 7.78 1.2 – 0.85 µs’[mm-1] GREY MATTER 2.64 – 2 0.62 – 0.43
* VALUES AVERAGED OVER THE WAVELENGTH RANGE MEASUREMENTS REALIZED FROM [650 – 900 nm]
A comparison between values of the optical properties shows that white
matter is more scattering than grey matter (µs’WHITE MATTER>>µs’GREY MATTER) for
both adult and neonatal brain, due to the high refractive index of the myelin
(lipid-rich) of the axons (van der Zee, 1992). The values of absorption and
scattering coefficients of the neonatal brain are smaller than the adult brain.
2.4.4.2 Functional and Anatomical areas of the brain
The cerebral cortex is responsible for conscious behaviour and contains
various functional areas: (a) the motor cortex is the area in the frontal lobe in
charge of movement and the area on each side of the brain is in charge of the
contralateral side of the body. Consequently activation of the right side of the
body produces a response on the left side of the brain and vice-versa. The
size of the area involved in the task depends on its complexity: the more
complex the task, the bigger the area involved; (b) the somatosensory cortex
is the region concerned with processing tactile and proprioceptive (position
sense) information; (c) the visual cortex is in charge of processing visual data,
Chapter 2. Fundamentals of Tissue Optics
22
and lies in the occipital lobe at the back of the head. This is highly specialized
for processing visual information from moving or static objects and for pattern
recognition; and (d) the auditory cortex is concerned with the hearing and lies
in the temporal lobe within the Sylvian fissure (Webster, 1992). The cerebral
cortex is further anatomically divided into the frontal, parietal, occipital, and
temporal lobes, named after the overlying bones of the skull and as shown in
figure 2.17.
Figure 2.17 Representation of the functional regions where are located the motor,
somatosensory, primary visual and auditory cortices (Webster, 1992) (p197), and the anatomically division of the brain (Marieb and Hoehn, 2006).
2.4.5 Summary
In the composition of biological tissue, static absorbers like water (and CSF),
melanin, and lipids, have fixed concentrations, and their contributions to the
overall attenuation are low within the therapeutic window. Although the CSF
layer is non-scattering its presence has been shown to affect light propagation
in the head (Okada and Delpy, 2003), (Okada et al, 1997), and it is sometimes
considered together with its surrounding membranes in light propagation
models (Fukui et al, 2003). However, the principal interaction of interest occurs
when light strikes a blood vessel. The light is absorbed by the dynamic
absorbers: oxyhaemoglobin (HbO2) and deoxyhaemoglobin (Hb). Changes in
the intensity of the incoming light can be converted into changes in
concentrations of oxyhaemoglobin and deoxyhaemoglobin, which can be used
to provide information on blood oxygenation status. Also, indirect information
can be gained from the different optical properties of the tissues, which are the
basis of several potential clinical applications, such as cerebral imaging
modality for mapping oxygenation and haemodynamics in the brain of
newborn infants or cortical functional activity in adults, (Hebden, 2003).
Chapter 2. Fundamentals of Tissue Optics
23
2.5 Optical monitoring of the brain injury in infants
2.5.1 Brain Injury during birth
Occasionally during the birth process, the baby may suffer a physical injury
that is simply the result of being born. This is sometimes called birth trauma or
birth injury. Premature babies (< 37 weeks of gestation) are more fragile and
may be more easily injured. The most common types of neurological problems
associated with birth injury in newborn babies are listed below and all manifest
themselves as disruption to the supply of blood and oxygen to vulnerable
areas of the brain (Gibson et al, 2005a):
I. Hypoxic ischemia encephalopathy (HIE) is a brain injury resulting from
(perinatal) asphyxia, and is one of the most commonly recognized causes of
severe, long-term neurological deficits in children, due to impaired CBF
(Shalak and Perlman, 2004). When the infant newborn brain is subject to the
asphyxia, its circulation becomes vasodilated with an increase of cerebral
blood volume (CBV) (Meek, 2002), (Meek et al, 1999 a), (Wyatt, 1993).
II. Periventricular leukomalacia (PVL) is the damage and softening of the
white matter around the ventricles (with subsequent cyst formation) due to an
incomplete state of development of the vascular supply and impairment in
regulation of cerebral blood flow (CBF) (Volpe, 2001). Studies in preterm
infants during the first 3 days of their life demonstrated an increased risk of
development of both IVH and PVL due to a low level of CBF (normal response
to extrauterine life) (Meek, 2002), (Meek et al, 1999 a).
III. Intraventricular haemorrhage (IVH) is a bleeding inside or around the
ventricles due to damage of capillaries occurring during periods of fluctuation
in blood pressure, which are common during birth (Gibson et al, 2005a),
(Whitelaw, 2001). The amount of bleeding varies and it is often described in
four grades: grades I (bleeding occurs just in a small area of the ventricles)
and II (bleeding also occurs inside the ventricles) are most common, and
grades III (ventricles are enlarged by the blood) and IV (bleeding into the brain
tissues around the ventricles) are the most serious and may result in long-
term brain injury to the baby (figure 2.18).
Chapter 2. Fundamentals of Tissue Optics
24
Figure 2.18 LEFT: Periventricular - intraventricular haemorrhage grades. Extracted from (Merestein et al, 1998) (p599), and RIGHT: Massive intraventricular haemorrhage, without distension of the ventricles (Whitelaw, 2001).
2.5.2 Monitoring brain injury
Currently, birth injuries have been diagnosed clinically by cranial ultrasound
(US- which gives only anatomical information), computerised tomography
(CT- rarely used on newborns because of the ionizing radiation and limited
due to high water content in the preterm infant brain (Halliday et al, 1998)),
and MRI (which is often not appropriate due to the fragile state of the infant
and the reluctance to transport the infant out of an intensive care unit) (Gibson
et al, 2005a). Other techniques such as electroencephalography (EEG),
magnetoencephalography (MEG), PET, and fMRI can be used to non-
invasively investigate cerebral function. Although these techniques are widely
used clinically, the instruments are expensive (with the exception of EEG),
require specially trained technical staff to operate, and patients have to be
moved from their controlled environment and taken to specially constructed
rooms where the investigation is carried out. In the case of PET, patients have
to be injected with radioisotopes, which is not appropriate for multiple
repeated studies. All of these factors make these imaging methods unsuitable
for imaging cerebral function in babies in real time, especially if they are being
cared for in an intensive care unit. It is desirable to have a clinical test that
would allow detection of potential brain injury before it occurs or as it is
Chapter 2. Fundamentals of Tissue Optics
25
occurring, when it is possible to intervene and reverse or prevent the brain
injury (Vainthianathan et al, 2004), (Hebden, 2003). Cerebral near-infrared
spectroscopy (NIRS) has the potential for immediate real-time assessment of
the adequacy of brain perfusion. NIRS is based on the physical principle that
oxygenated and deoxygenated haemoglobin have different light-transmitting
characteristics in the near-infrared region. Thus, a light-emitting/detecting
system can be used to assess the relative amount of oxygen saturated
haemoglobin in the tissue being sampled or adapted as a cotside instrument
for continuous monitoring (Cope and Delpy, 1988).
2.6 Modelling of photon transport in tissue
An understanding of the propagation of light in tissue enables optical
techniques, such as NIR spectroscopy to yield quantitative information about
tissue oxygenation and haemodynamics, and NIR imaging to produce surface
maps of the cortex activation or volume images of spatial/temporal changes in
[HbO2], [Hb] and [total Hb] in the infant brain.
2.6.1 Modified Beer-Lambert Law (MBLL)
Scattering causes light to travel extra distance in tissue, increasing the
probability of photon absorption. The differential pathlength DP, the real
optical pathlength, can be obtained from the differential pathlength factor DPF,
given by:
d.DPFDP = [2.13]
where the geometrical distance between the source and detector is d. The
differential pathlength factor or scaling factor will depend on the number of
scattering events that occur. The DPF will in practise be a function of the
scattering coefficient (it will increase with increasing µs), the anisotropy g, the
absorption coefficient (it will decrease with increasing µa), and the geometry of
the medium. It must be included in the Beer-Lambert law to describe
attenuation in a scattering medium (Matcher et al, 1993). The DPF can be
considered approximately constant for a given tissue, since the measured
Chapter 2. Fundamentals of Tissue Optics
26
difference in attenuation is small compared with the large constant
background attenuation in tissue (Elwell, 1995). It is also necessary to
introduce an additive term G due to the scattering losses. Thus, )Gd.DPF.(
)d(ae.II +−= µ
0 [2.14]
This is known as the modified Beer-Lambert Law. G is dependent on the
measurement geometry and the scattering coefficient of the tissue under
study and is largely unknown. Consequently, spectroscopic measurements
generally assume that G is constant during the measurement period and
attempt to quantify changes in the absorption instead of absolute values:
==
=−= 21
21
1221 µ∆∆ .d.DPFIIlogAAA e
2112 ]C[).(.d.DPFAA ∆λα=−
[2.15]
The differences in attenuation measured between two oxygenation states is
given by ∆A21 corresponding to an absorption change of ∆µa21. The DPF can
be measured by two methods: intensity modulated optical spectroscopy (see
section 3.10.2) or time of flight (see section 3.10.3). According to Duncan et al
(1995), the values of the differential pathlength factor are 4.99 (± 9%) for
infant head and 6.26 (± 14.1%) for adult head at 807 nm.
2.6.1.1 Determination of the blood oxygenation status with NIR light
The parameter which establishes the degree of the oxygenation of blood is
the oxygen saturation (SO2) and it is calculated using the absolute values of
concentration and is given by:
1002
22 .
]Hb[]HbO[]HbO[[%]SO
+=
[2.16]
Considering the contribution of oxy and deoxy-haemoglobin (equation [2.7]), it
is possible to determine the absorption coefficient of blood at two different λ’s
as follows:
]).[(]).[( 1211 2HbHbO HbHbO λαλαµλ += [2.17]
]).[(]).[( 2222 2HbHbO HbHbO λαλαµλ += [2.18]
Chapter 2. Fundamentals of Tissue Optics
27
By solving the simultaneous equations [2.16], [2.17] and [2.18], it is possible
to obtain the oxygen saturation as shown in equation [2.19].
100.).().().().(
).().([%]
22212
12
11221
212
λλλλ
λλ
µλαµλαµλαµλαµλαµλα
HbOHbOHbHb
HbHbSO−+−
−=
[2.19]
Figure 2.9 provides the values of αHbO2, αHb for any two wavelengths in the
NIR range, so by measuring the µλ1 and µλ2, the oxygen saturation can be
determined by using equation [2.19]. SO2 is directly related to the supply of
blood and usage by the tissue, and indicates its functional activity. For
instance, non-invasive measurement of SO2 is a common bedside procedure
in hospitals. It is performed continuously, safely and instantaneously by pulse
oximeters. The measurements are typically made on fingers and/or ear lobes.
A two-wavelength method is used to quantify changes in light attenuation
during the systolic phase of blood flow in tissue, which are converted to levels
of oxygen blood saturation. For more details about pulse oximetry see
Mendelson (1992).
2.6.1.2 Determination of the chromophores concentrations
Because the two forms of haemoglobin have different absorption spectra in
the NIR range, it is possible to measure the relative concentration of
oxyhaemoglobin ([HbO2]) and deoxyhaemoglobin ([Hb]) using measurements
at two wavelengths (λ1,λ2) of the differences in attenuation or absorption
(∆A(λ1),∆A(λ2)). If just the contribution of two chromophores is considered, the
equations [2.7] and [2.15] can be written as follows:
( )]Hb[).(]HbO[).(.d).(DPF)(A HbHbO ∆λα∆λαλλ∆ 12111 2+= [2.20]
( )]Hb[).(]HbO[).(.d).(DPF)(A HbHbO ∆λα∆λαλλ∆ 22222 2+= [2.21]
In order to determine the changes in concentrations of the two states of
oxygenation, the equations [2.22] and [2.23] are rearranged. These values
can describe how well the blood is oxygenated.
Chapter 2. Fundamentals of Tissue Optics
28
d)).().()().(()(DPF
)(A).()(DPF
)(A).(]HbO[
HbOHbHbOHb
HbOHbO
1221
2
21
1
12
222
22
λαλαλαλαλλ∆λα
λλ∆λα
∆−
−=
[2.22]
d)).().()().(()(DPF
)(A).()(DPF
)(A).(]Hb[
HbOHbHbOHb
HbHb
1221
1
12
2
21
22λαλαλαλαλλ∆
λαλλ∆
λα∆
−
−=
[2.23]
2.6.2 The Radiative Transfer Equation (RTE)
In the RTE approach light is treated as composed of distinct particles
(photons) propagating through a medium. The model is restricted to
interactions between light particles themselves and is derived by considering
changes in energy flow due to incoming, outgoing, absorbed and emitted
photons within an infinitesimal volume dV in the medium (energy balance).
The model considers a small packet of light energy defined by its position r,
direction of propagation ŝ, over a time interval dt, and with propagation speed
c (figure 2.19).
Figure 2.19 Model for RTE
The change in energy radiance I(r,t,ŝ) is equal to the loss in energy due to
absorption and scattering out of ŝ, plus the gains in energy from light
scattered into the ŝ-directed packet from other directions and from any local
source of the light at r. This energy balance is represented by the individual
terms in the RTE:
=∇•+∂
∂ ),,(),,(1 strIst
strIc
)))
),,(').',,().',(')],,().'[(4
2 strqsdstrIsspstrI ssa))))))
+++−= ∫π
µµµ
[2.24]
Each term in equation [2.24] represents in time domain:
direction ŝ’
PHOTONS OUT (Iout) PHOTONS IN (Iin)
PHOTONS SCATTERED FROM ANOTHER DIRECTION TO DIRECTION OF INTEREST PHOTONS SCATTERED TO
ANOTHER DIRECTION
PHOTONS ABSORBED
dV r
x
z y
Chapter 2. Fundamentals of Tissue Optics
29
),,(),,(1 strIs
tstrI
c))
)
∇•+∂
∂
The difference between the number of photons entering the volume and the number of photons leaving it per
unit time
),,().'( strIsa)
µµ +
The attenuation given to light due to absorption and scattering
∫π
µ4
2 ').',,().',(' sdstrIssps))))
The increase in the light due to
scatter from all directions to final direction ŝ’
),,( strq)
Local sources [W/m3/sr]
(for instance, fluorescence)
Two important parameters are Φ(r,t) which represents photon density or
diffuse photon fluence (inside the element), and J(r,t) which is the photon flux
or current (at its boundary). The latter is a measurable parameter and allows
equation [2.24] to be solved for µs’ and µa, respectively (Kaltenbach and
Kaschke, 1993).
∫=π
Φ4
').',,(),( sdstrItr))
[2.25]
∫=π4
').',,(.),( sdstrIstrJ)))
[2.26]
Exact solutions for the RTE exist for simple cases such as isotropic scattering
in simple geometries. A more in-depth treatment of the subject is given in the
review papers by (Arridge and Hebden, 1997) and (Patterson et al, 1991).
2.6.2.1 The diffusion approximation to the RTE
Three variables in the RTE depend on direction ŝ: the radiance I(r,t,ŝ), the
phase function p(ŝ,ŝ’) and the source term q(r,t,ŝ). If these are expanded into
spherical harmonics, an infinite series of equations which approximate to the
RTE is obtained. The PN approximation is obtained by taking the first N
spherical harmonics, of which the simplest is the time-dependent P1
approximation. If the following assumptions are made: (a) scatter is the
dominant interaction: µs’ >> µa, (b) phase function p(ŝ,ŝ’) is independent of the
Photons out
Photons scattered to
Photons absorbed in the volume dV
dV
Photons in
dV
another direction
direction ŝ’
dV
Photons scattered from
another direction to direction of interest
Local sourcesdirection ŝ’
Chapter 2. Fundamentals of Tissue Optics
30
absolute angle, (c) photon flux J(r,t) changes slowly dJ(r,t)/dt = 0 and (d) all
sources are isotropic, the result is the time-dependent Diffusion equation:
)t,r(q)t,r()r()t,r()r(t
)t,r(c a =+∇•∇−
∂∂ ΦµΦκΦ1
[2.27]
where:
)t,r()r()t,r(J Φκ ∇−= (1st Fick’s Law) [2.28]
κ(r) is the diffusion coefficient defined as:
)](')(.[31)(
rrr
sa µµκ
+=
[2.29]
The diffusion equation [2.27] has been widely used to model light transport in
tissue, although it is necessary to assume that light propagates diffusively
(which is generally the case in bulk tissue), and the source and detector are
separated in space and time, to ensure that the light is diffuse when it reaches
the detector. By contrast, these assumptions generally do not hold near the
source, near the surface and internal boundaries and in anisotropic tissues,
and in regions of either high absorption or low scatter (voids regions, such as
the ventricles and CSF). For these cases, higher order approximations to the
RTE may be required (Gibson et al, 2005a), and models which incorporate
void regions within a diffusing model (Riley et al, 2000).
2.6.2.1.1 Green’s functions
A general method for solving partial differential equations (PDE) such as the
diffusion equation is the application of Green’s functions, where the source
term q(r,t) consists of an infinitely short pulse or δ-function (any other source
can be obtained by convolution (Arridge et al, 1992)). An example of analytical
solution was developed by Patterson et al (1989) for a homogeneous slab
where boundaries are described using the so-called method of images. This
yields time-resolved reflectance and transmittance equations, and least-
squared fitting to experimental data allows the optical coefficients to be
determined. For instance, during the determination of the optical properties of
a liquid slab of intralipid solution (mixture used for reference phantoms, see
section 5.2.2.2), a rectangular transparent receptacle is filled with the
Chapter 2. Fundamentals of Tissue Optics
31
homogenous solution (nominal properties of the mixture are µs’ = 1 mm-1,
µa = 0.01 mm-1). A connector for a light source is attached on one of the
faces, and another for a detector is attached on the opposite face. Both
connectors are at the same height and located in the middle of each face. A
representation of the experiment is shown in figure 2.20. A short pulse from a
laser source (see section 4.2.1) is applied and the time-resolved transmitted
intensity across the solution is detected (50 mm of thickness). The values of
the absorption and scattering coefficients are obtained from the temporal
distribution of transmitted light due by fitting with the corresponding Green
function (figure 2.20). For more details see (Patterson et al, 1989).
time (ps)
0 1000 2000 3000 4000 5000 6000 7000 8000
Rel
ativ
e Tr
ansm
ittan
ce
0.0
0.2
0.4
0.6
0.8
1.0
Temporal distribution of transmitted light
Fitted data curve
µa = 9.344135e-03 +/- 9.383923 10-6
mm-1
µs'= 0.982167 +/- 0.000561 mm-1
χ2= 1.020439
Figure 2.20 TOP: A measured transmittance signal and the correspondent fitted curve,
BOTTOM: A measured transmittance signal and the correspondent fitted curve.
Other analytical solutions for simple geometries, such as spheres and
cylinders, as well as the frequency domain equivalents of the equations, are
given in (Arridge et al, 1992). For more complex geometries, the solutions are
solved numerically.
2.6.2.1.2 Finite element method (FEM)
The finite element method can be applied in order to solve numerically the
partial differential equations (such as diffusion equation) for arbitrary and
Transparent recipient
50 mm
TRANSMITTED LIGHT
THE APPLIED LIGHT FROM
A LASER SOURCE
Intralipid solution
Chapter 2. Fundamentals of Tissue Optics
32
complex geometries. Arridge et al (1993) stated that FEM can be applied
anywhere where a differential equation formulation is available for the
transport model. The method involves dividing a region of interest (or domain)
into a finite number of volume or area elements. The boundary of one element
consists of discrete points called nodes (or connecting points). Surface
domains may be subdivided into triangles and volumes may be subdivided
into tetrahedral (polyhedrons) shapes. Each element has its individual set of
optical properties (µa and µs’). The shape and distribution of the elements is
ideally defined by automatic meshing algorithms (e.g. NETGEN developed by
Schöberl, (1997)). Figure 2.21 shows a finite element (volume) mesh
generated using automatic meshing algorithms (NETGEN).
MESH CHARACTERISTICS
i. 404 POINTS OR NODES
ii. 1459 ELEMENTS (TETRAHEDRAL)
iii. 546 SURFACE ELEMENTS (TRIANGLES)
Figure 2.21 A head-shaped finite element mesh (left) and the surface cut away to show internal structures (right).
The photon density approximation Φi(t) is then calculated at each node i, and
summed over all N nodes in the region. It can be expressed using the
piecewise polynomial function:
∑=N
iii
h )r(u)t()t,r( ΦΦ [2.30]
The h superscript denotes the iterative process that is calculated at the nodes
and which provides an approximate solution for photon density Φ(t). The ui(r)
is a basis function and it describes the way that the function Φi(t) is allowed to
vary over an element (between nodes). Its simplest form is to be a constant,
although a variety of basis functions can be used (Arridge and Schweiger,
1993). The advantages of this method are its computational speed, the high
flexibility (as for the Monte Carlo method) when applied to complex
geometries and it can model photon density and flux everywhere. The
Chapter 2. Fundamentals of Tissue Optics
33
disadvantages are that there is no means of deriving individual photon
histories and it is subject to the same limitations as the diffusion equation
(Arridge et al, 1993).
2.6.2.2 The Monte Carlo Method
Many problems of practical interest often involve a range of light sources,
multiple tissues types, and complex geometries. Analytical solutions for
realistic scenarios are complicated, even when possible. These more realistic
cases are instead solved with numerical techniques. For RTE problems, the
most widely used approach is the Monte Carlo (MC) method. In this approach
to photon transport, single photons are traced through the sample step by
step, and the distribution of light is built up from these single-photon
trajectories. The parameters of each step are calculated using functions
whose arguments are random numbers (Vo-Dinh, 2003). The advantages of
the MC method are simple implementation, the ability to handle any complex
geometries and inhomogeneous media (Patterson et al, 1991). The main
disadvantage is the associated high computational cost, since the statistical
accuracy of results is proportional to [number of traced photons]½. As the
numbers of photons in the simulation grows toward infinity, the MC prediction
for the light distribution approaches an exact solution of the RTE (Niemz,
1999). A MC simulation of absorption and scattering has 5 main steps: source
photon generation, pathway generation, absorption, elimination, and
detection.
2.6.3 Optical image reconstruction
Optical image reconstruction involves deducing the internal distribution of µa
and µs’ that correspond to a set of measurements recorded. The process
applied to optical tomography involves the solution of the forward and the
inverse problems. The forward problem is to predict the distribution of light y in
the object under examination. This uses a model of photon migration in the
object to generate a sensitivity matrix J (or forward operator) which relates the
Chapter 2. Fundamentals of Tissue Optics
34
distribution of light on the boundary of the object with its internal properties x,
as shown in equation [2.31].
x.Jy = [2.31]
The forward problem can generate model data for comparison with
experimental data or to test reconstruction techniques. The distribution of light
y can be determined with previous knowledge of the geometry of the object,
the location of the sources and detectors, and background parameters. In a
similar way, in the inverse problem an image (2D or 3D) of the internal
properties of the object can be recovered from the distribution of
measurements on the boundary y by inverting the forward operator J, as
shown in equation [2.32].
yJx .1−= [2.32]
However, this inversion is ill posed (the solution may not be unique) and is
generally highly underdetermined (the number of unknowns exceeds the
amount of data), and does not yield to straightforward analytical methods
(Arridge and Hebden, 1997). Image reconstruction methods can be
categorised as either linear and or non-linear.
2.6.3.1 Linear reconstruction
The linear reconstruction method is the simplest way to recover an optical
image. This involves considering changes in the optical properties ∆x = (x-x0),
and corresponding changes in the measured data ∆y = (y-y0). By expanding
equation in terms of the Taylor series yields:
...)xx(F)xx)(x(Fyy ''' +−+−+= 20000 [2.33]
where F’, F’’ … are the first derivative, second derivative and so on. In the
case of difference imaging, which is usually employed (see section 4.2.7.5),
the changes in the optical properties ∆x are small, and equation [2.33] can be
linearized, yielding:
x.Jy ∆=∆ [2.34]
Chapter 2. Fundamentals of Tissue Optics
35
As described earlier, the image reconstruction consists of the product of the
inversion of the forward operator J and the change in measurements or
difference between two states (∆y):
y.Jx ∆∆ 1−= [2.35]
There are several techniques to invert a matrix; one of the most common is
the Tikhonov regularisation of the Moore-Penrose generalised inverse
method:
y.)I.J.J(Jx TT ∆λ∆ 1−+= [2.36]
where the parameter of regularisation λ is typically ~0.01 and I is the identity
matrix. An advantage of linear reconstruction is that it does not require a
good forward model for good results to be achieved, although this method can
only reconstruct small changes in the optical properties, otherwise the
reconstruction will probably fail, especially if the reference phantom has
optical properties that are not close to those of the tissue analized. For more
details see (Gibson et al, 2005a) and (Gibson et al, 2005b).
2.6.3.2 Non-linear reconstruction
Non-linear reconstruction is a method where the difference between the
values calculated by a forward model and the experimental data is used to
update the sensitivity matrix J of the model and to minimise this difference
between the estimated and the measured data. The image reconstruction
package developed at UCL, known as TOAST (Temporal Optical Absorption
and Scattering Tomography) is a software for image reconstruction in diffuse
optical tomography, which involves:
• a forward solver, a finite element model (FEM) to simulate the light
transport in tissue (scattering medium), which has the flexibility to handle
complex geometries (essential for imaging a real infant’s brain); and
• an iterative inverse solver, a non-linear reconstruction algorithm which
compares the measured TPSF data and the data generated by the FEM
model (forward solver), acting iteratively on the trial solution. Once the
Chapter 2. Fundamentals of Tissue Optics
36
difference is below a set limit, the generated model is considered a good
approximation with the true tissue optical properties.
A more detailed description of the method of reconstruction and the main
theoretical principles involved in TOAST are given elsewhere (Arridge, 1999),
(Arridge, 1993).